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Curves of given prank with trivial automorphism group
, 2007
"... Let k be an algebraically closed field of characteristic p> 0. Suppose g ≥ 3 and 0 ≤ f ≤ g. We prove there is a smooth projective kcurve of genus g and prank f with no nontrivial automorphisms. In addition, we prove there is a smooth projective hyperelliptic kcurve of genus g and prank f w ..."
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Cited by 6 (5 self)
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Let k be an algebraically closed field of characteristic p> 0. Suppose g ≥ 3 and 0 ≤ f ≤ g. We prove there is a smooth projective kcurve of genus g and prank f with no nontrivial automorphisms. In addition, we prove there is a smooth projective hyperelliptic kcurve of genus g and prank f whose only nontrivial automorphism is the hyperelliptic involution. The proof involves computations about the dimension of the moduli space of (hyperelliptic) kcurves of genus g and prank f with extra automorphisms.
Random Dieudonné modules, random pdivisible groups, and random curves over finite fields
, 2012
"... We describe a probability distribution on isomorphism classes of principally quasipolarized pdivisible groups over a finite field k of characteristic p which can reasonably be thought of as “uniform distribution, ” and we compute the distribution of various statistics (pcorank, anumber, etc.) of ..."
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We describe a probability distribution on isomorphism classes of principally quasipolarized pdivisible groups over a finite field k of characteristic p which can reasonably be thought of as “uniform distribution, ” and we compute the distribution of various statistics (pcorank, anumber, etc.) of pdivisible groups drawn from this distribution. It is then natural to ask to what extent the pdivisible groups attached to a randomly chosen hyperelliptic curve (resp. curve, resp. abelian variety) over k are uniformly distributed in this sense. This heuristic is analogous to conjectures of CohenLenstra type for char k = p, in which case the random pdivisible group is defined by a random matrix recording the action of Frobenius. Extensive numerical investigation reveals some cases of agreement with the heuristic and some interesting discrepancies. For example, plane curves over F3 appear substantially less likely to be ordinary than hyperelliptic curves over F3. 1
Generic Newton polygons for curves of given prank
 Algebraic Curves and Finite Fields: Cryptography and Other Applications, volume 16 of RICAM
, 2014
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